Problem

Part I Multiple Choice

4. Let

$$f(x,y)= \begin{cases} \dfrac{x\sin y^2}{1-e^{x^2+y^4}}, & \text{if }(x,y)\ne(0,0), \\ 0, & \text{if }(x,y)=(0,0). \end{cases}$$

(a) $\lim_{x\to 0}\lim_{y\to 0} f(x,y)=0$.

(b) $\lim_{x\to 0}\lim_{y\to 0} f(x,y)=\lim_{y\to 0}\lim_{x\to 0} f(x,y)$.

(c) $\lim_{(x,y)\to(0,0)}f(x,y)=0$.

Among the above three statements, how many of them are true?

A) Only one.

B) Only two.

C) All of them.

D) None of them.